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Accurate measurements on the behaviour of a climbing rope during the entire fall can be explained if the undamped harmonic oscillator is complemented by a non-linear term up to the maximum impact force, and then, near the maximum force in the rope, internal friction in the rope is added that ensures the rapid relaxation of the rope to its rest position.
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formula assures that the transformation will continue to be valid for different gravity fields, as over a slope making less than 90 degrees with the horizontal. This simple undamped harmonic oscillator model of a rope, however, does not correctly describe the entire fall process of real ropes.
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to 10 feet below—a factor 2 fall). This fall produces far more force on the climber and the gear than if a similar 20 foot fall had occurred 100 feet above the belayer. In the latter case (a fall factor of 0.2), the rope acts like a bigger, longer rubber band, and its stretch more effectively
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falls past the belayer (two times the distance of the rope length between them), or the anchor if the climber is solo climbing the route using a self-belay. As soon as the climber clips the rope into protection above the belay, the fall factor drops below 2.
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The impact force is defined as the maximum tension in the rope when a climber falls. We first state an equation for this quantity and describe its interpretation, and then show its derivation and how it can be put into a more convenient form.
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between the rope and particularly the last clipped carabiner. "Dry" friction (i.e., a frictional force that is velocity-independent) leads to an effective rope length smaller than the available length
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As a numerical example, consider a fall of 20 feet that occurs with 10 feet of rope out (i.e., the climber has placed no protection and falls from 10 feet above the
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The smallest possible fall factor is zero. This occurs, for example, in top-rope a fall onto a rope with no slack. The rope stretches, so although
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There are two factors of two involved in the interpretation of this equation. First, the maximum force on the top piece of protection is roughly 2
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is short and fixed, while the distance the climber can fall depends on the gaps between anchor points of the safety cable (i.e. the climber's
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When climbing from the ground up, the maximum possible fall factor is 1, since any greater fall would mean that the climber hit the ground.
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Other than fixed properties of the system, this form of the equation shows that the impact force depends only on the fall factor.
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906:{\displaystyle F_{max}=mg+{\sqrt {(mg)^{2}+F_{0}(F_{0}-2m_{0}g_{0}){\frac {m}{m_{0}}}{\frac {g}{g_{0}}}{\frac {f}{f_{0}}}}}}
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will fall down the safety cable until it reaches an anchor point); to mitigate this, via ferrata climbers can use
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We will see below that when varying the height of the fall while keeping the fall factor fixed, the quantity
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It is the main factor determining the violence of the forces acting on the climber and the gear.
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used in the fall is 80 kg. Using these values to eliminate the unknown quantity
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leads to an expression of the impact force as a function of arbitrary fall heights
53:) a climber falls before the climber's rope begins to stretch and the rope length (
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of a given rope. However, rope manufacturers give only the rope’s impact force
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can be as high as 2. This can occur only when a lead climber who has placed no
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499:{\displaystyle mgh={\frac {1}{2}}kx_{max}^{2}-mgx_{max}\ ;\ F_{max}=kx_{max}.}
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When the rope is clipped into several carabiners between the climber and the
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which is a property of the material that the rope is constructed from. Here
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and its static and dynamic elongations that are measured under standard
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1043:"Physics of climbing ropes: impact forces, fall factors and rope drag"
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is the spring constant of the portion of the rope that is in play.
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1010:"The physics of a climbing rope under a heavy dynamic load"
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its cross-sectional area. Solution of the quadratic gives
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Federación Española de
Deportes de Montaña y Escalada
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624:{\displaystyle F_{max}=mg+{\sqrt {(mg)^{2}+2mgEqf}}.}
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517:. It is convenient to express things in terms of the
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International
Climbing and Mountaineering Federation
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Fédération française de la montagne et de l'escalade
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Conservation of energy at rope's maximum elongation
284:{\displaystyle F_{max}=mg+{\sqrt {(mg)^{2}+2mghk}},}
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Equation for the impact force and its interpretation
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667:of 2 × 2.3 m with an available rope length
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1087:"Climbing Physics - Understanding Fall Factors"
57:) available to absorb the energy of the fall,
16:Mathematical ratio relevant to climbing safety
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1117:Contains full derivation of equation in Notes
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1014:Journal of SPORTS ENGINEERING AND TECHNOLOGY
23:The climber will fall about the same height
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1070:"The Standard Equation for Impact Force"
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1008:Leuthäusser, Ulrich (June 17, 2016).
955:and thus increases the impact force.
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927:" based on UIAA test into the above
509:The maximum force on the climber is
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1103:"UKC - Understanding fall factors"
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1113:"Rock Climbing Fall Impact Force"
89:{\displaystyle f={\frac {h}{L}}.}
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1489:IFSC World Championships winners
985:"Get into via ferrata: the gear"
983:Davies, Carey (July 16, 2017).
660:fall conditions: A fall height
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674:= 2.6m leads to a fall factor
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692:= 1.77 and a fall velocity
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354:Derivation of the equation
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728:, arbitrary fall factors
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2387:Mountain Equipment Co-op
1022:10.1177/1754337116651184
923:from the derivation of "
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155:In falls occurring on a
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2317:Eastern Mountain Sports
2183:Canadian Alpine Journal
2176:American Alpine Journal
1439:Roof and tunnel hacking
302:is the fall height and
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2154:Magazines and journals
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192:(HO) the impact force
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2377:Marmot Mountain Works
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137:multi-pitch climbing
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1119:. vCalc. 2014-04-11
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2546:
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2543:
2541:International
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2500:
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2441:Organizations
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2312:Early Winters
2310:
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2300:
2298:
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2293:
2292:Black Diamond
2290:
2289:
2287:
2283:
2273:
2272:
2268:
2266:
2265:
2261:
2259:
2258:
2257:The Dawn Wall
2254:
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2251:
2247:
2245:
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2240:
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2123:
2120:
2118:
2117:Bachar ladder
2115:
2114:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2067:Climbing wall
2065:
2063:
2060:
2058:
2055:
2054:
2053:
2050:
2048:
2045:
2043:
2042:Approach shoe
2040:
2038:
2035:
2034:
2032:
2028:
2020:
2017:
2016:
2015:
2012:
2010:
2007:
2003:
2000:
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1995:
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1990:
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1662:Climbing rope
1660:
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1543:First ascents
1541:
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1536:
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1177:Rock climbing
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782:
779:
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768:
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762:
757:
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747:
739:
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736:of the form:
735:
731:
727:
723:
719:
712:
705:
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680:
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666:
659:
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648:
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618:
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200:
198:
191:
183:
181:
174:
172:
170:
166:
162:
158:
153:
150:
146:
145:lead climbing
142:
138:
133:
130:
128:
121:
117:
110:
108:
105:
100:
83:
78:
75:
70:
67:
60:
59:
58:
56:
52:
48:
47:
42:
38:
34:
33:lead climbing
26:
21:
2624:
2612:
2600:
2588:
2532:USA Climbing
2432:Wild Country
2427:Therm-a-Rest
2269:
2262:
2255:
2248:
2243:The Alpinist
2241:
2223:
2216:
2209:
2202:
2195:
2188:
2181:
2174:
2167:
2160:
2122:Campus board
2098:
2077:First ascent
1821:Ice and snow
1618:Belay device
1469:Alpine clubs
1429:Ropes course
1342:Alpine style
1121:. Retrieved
1116:
1090:. Retrieved
1073:. Retrieved
1049:. Retrieved
1036:
1025:. Retrieved
1013:
1003:
992:. Retrieved
988:
978:
952:
942:
928:
924:
917:
915:
733:
729:
725:
721:
714:
707:
700:
693:
682:
675:
668:
661:
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134:
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126:
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101:
98:
54:
50:
45:
44:
40:
37:dynamic rope
30:
24:
2626:WikiProject
2462:Alpine Club
2457:Access Fund
2367:Lowe Alpine
2362:La Sportiva
2099:Fall factor
1992:Self-arrest
1968:Redpointing
1891:Snow anchor
1645:Boulder mat
1594:Daisy chain
1564:Terminology
1555:Terminology
1434:Rooftopping
1376:Hillwalking
1364:Dry-tooling
1330:Via ferrata
1243:Traditional
1221:Competition
1202:Multi-pitch
1185:Disciplines
713:. The mass
327:, we have
157:via ferrata
41:fall factor
2642:Categories
2250:El Capitan
2014:Traversing
1963:Onsighting
1943:Glissading
1926:Dülfersitz
1921:Australian
1884:Transeiver
1829:Alpenstock
1762:Portaledge
1750:Copperhead
1689:Descenders
1623:Auto belay
1414:Commercial
1394:Buildering
1371:Scrambling
1281:Rock types
1272:Top roping
1209:Bouldering
1123:2014-04-11
1092:2008-06-14
1075:2009-04-17
1051:2011-01-15
1027:2016-06-29
994:2019-02-16
971:References
149:protection
2653:Mechanics
2402:Patagonia
2285:Companies
2271:Hard Grit
2264:Free Solo
2132:MoonBoard
2127:Hangboard
2113:Training
2062:Guidebook
2037:Ape index
2002:Rope team
1916:Abseiling
1861:Ice screw
1767:Quickdraw
1650:Carabiner
1572:Equipment
1550:Technique
1516:Equipment
1399:Canyoning
1337:Himalayan
1265:Rope solo
1255:Free solo
822:−
422:−
161:carabiner
2648:Climbing
2602:Category
2449:National
2347:JanSport
2322:Five Ten
2197:Desnivel
2190:Climbing
2169:Alpinist
2094:Exposure
2072:Chipping
2019:Tyrolean
2009:Spotting
1980:Crevasse
1958:Flashing
1948:Jumaring
1933:Belaying
1896:Snowshoe
1854:Ice tool
1844:Crampons
1694:Figure-8
1606:Ascender
1474:Climbers
1214:Highball
1197:Big wall
1162:Climbing
959:See also
949:friction
35:using a
2614:Commons
2568:(UIMLA)
2562:(UIAGM)
1953:Leading
1870:Rescue
1849:Ice axe
1799:Camalot
1733:Lanyard
1716:Skyhook
1701:Harness
1672:Dynamic
1655:Maillon
1628:Glasses
1509:Everest
1500:Deaths
1419:Parkour
965:Whipper
945:belayer
165:lanyard
104:belayer
2658:Ratios
2590:Portal
2556:(UIAA)
2550:(IFSC)
2382:Millet
2372:Mammut
2337:Grivel
2307:Deuter
2225:Summit
2109:Guides
2104:Grades
1975:Rescue
1909:Action
1811:Tricam
1777:Grigri
1723:Hammer
1677:Static
1633:Gloves
1601:Anchor
1521:Brands
1325:Alpine
453:
447:
365:gives
294:where
39:, the
2407:Petzl
2357:Kelty
2147:Media
2082:Pitch
2047:Clean
2030:Other
1879:RECCO
1806:Shoes
1794:SLCDs
1789:Sling
1757:Piton
1706:Hooks
1684:Chalk
1611:Croll
1589:Aider
1526:Knots
1462:Lists
1444:Slide
1409:Grass
1404:Crane
1387:Other
1359:Mixed
1288:Crack
1238:Sport
1226:Speed
1169:Types
1046:(PDF)
526:k L/q
139:(and
2297:CAMP
2087:Topo
2057:Beta
1985:Self
1782:Revo
1772:SLDs
1711:Fifi
1667:Cord
1640:Bolt
1581:Rock
1449:Tree
1424:Pole
1303:Slab
1298:Roof
1293:Face
1250:Solo
1233:Free
658:UIAA
2417:REI
2412:Rab
1745:Nut
1738:PAC
1728:Hex
1354:Ice
1192:Aid
1018:doi
931:max
701:2gh
699:= (
515:-mg
513:max
362:max
331:max
325:f=0
320:max
196:max
135:In
31:In
2644::
1115:.
1016:.
1012:.
987:.
925:Eq
687:/L
681:=
643:mg
524:=
350:.
348:mg
344:mg
340:mg
336:mg
334:=2
311:hk
296:mg
171:.
1154:e
1147:t
1140:v
1126:.
1105:.
1095:.
1078:.
1054:.
1030:.
1020::
997:.
953:L
929:F
921:0
918:g
895:0
891:f
887:f
878:0
874:g
870:g
861:0
857:m
853:m
848:)
843:0
839:g
833:0
829:m
825:2
817:0
813:F
809:(
804:0
800:F
796:+
791:2
787:)
783:g
780:m
777:(
772:+
769:g
766:m
763:=
758:x
755:a
752:m
748:F
734:g
730:f
726:h
722:E
717:0
715:m
710:0
708:h
703:0
696:0
694:v
689:0
685:0
683:h
678:0
676:f
671:0
669:L
664:0
662:h
653:0
651:F
647:E
639:h
619:.
614:f
611:q
608:E
605:g
602:m
599:2
596:+
591:2
587:)
583:g
580:m
577:(
572:+
569:g
566:m
563:=
558:x
555:a
552:m
548:F
534:q
530:L
522:E
511:F
494:.
489:x
486:a
483:m
479:x
475:k
472:=
467:x
464:a
461:m
457:F
450:;
442:x
439:a
436:m
432:x
428:g
425:m
417:2
412:x
409:a
406:m
402:x
398:k
393:2
390:1
385:=
382:h
379:g
376:m
360:x
329:F
318:F
304:k
300:h
279:,
274:k
271:h
268:g
265:m
262:2
259:+
254:2
250:)
246:g
243:m
240:(
235:+
232:g
229:m
226:=
221:x
218:a
215:m
211:F
194:F
127:h
84:.
79:L
76:h
71:=
68:f
55:L
51:h
46:f
43:(
25:h
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
